Naming Platonic solids, counting faces and vertices, and working with nets.
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What gets asked
- Name a Platonic solid from its faces.
- Count the faces, vertices and edges of a solid.
- Use the relationship between vertices, edges and faces to find a missing count.
- Match a net to the solid it folds into.
You must be able to
- Name a Platonic solid by its number of faces, not its corners.
- Count vertices, edges and faces without missing any hidden ones.
- Use to find a missing count for a polyhedron.
- Check that a net will fold up without any faces overlapping.
Traps that cost marks
- Naming a solid by counting its vertices instead of its faces. A Platonic solid takes its name from its number of faces, not its corners.
- Counting only the faces you can see in a drawing of a solid. A drawing hides some faces. Picture the whole object, or use its net, to count them all.
- Putting the number of edges in the spot where vertices belong in . Keep , and straight: vertices, edges, then faces, in that order.
- Assuming any arrangement of the right faces makes a valid net. Picture folding the net up. The faces must meet edge to edge without overlapping.
Worked example
- Substitute: .
- Simplify: .
- .
Polyhedra and nets
Only five solids in the whole world are perfectly regular, and there can never be a sixth.
- polyhedron
- A solid whose faces are all flat straight-sided shapes. A ball is not a polyhedron.
- face
- One of the flat surfaces of a solid.
- edge
- The straight line where two faces of a solid meet.
- vertex
- A corner of a solid, where three or more edges meet. Two or more of them are called vertices.
- Platonic solid
- A polyhedron whose faces are all the same shape, with equal sides and equal angles. The same number of faces meet at every vertex.
- net
- The flat shape you get by unfolding a solid so that every face lies open.
A solid with flat faces is a polyhedron. Three things get counted: its faces, its edges and its vertices. An edge is where two faces meet. Three or more edges meet at every vertex. A drawing hides the back of a solid, so picture the whole object, or use its net.
Only five polyhedra are perfectly regular. Each one takes its name from its number of faces, and those numbers are , , , and . These are the Platonic solids. The one with triangles is the tetrahedron, and the one with squares is the cube.
Count the faces, edges and vertices of any polyhedron and one sum always comes out at . A cube gives . A tetrahedron gives . So when two of the three counts are known, the third can be worked out.
A net has to fold up. Having the right faces is not enough on its own: they must join edge to edge, with no overlap. A square pyramid needs triangles and its square base as well. Leave the base out and the net folds into an open shell.
Rules to remember
- for every polyhedron
Examples
Worked answer
- A cube has square faces.
- Each face has edges, but two faces share every edge: .
- Now use the rule: , so .
- A cube does have corners.
Answer: faces, edges and vertices
Sharing halves the edge count, and the rule then gives the vertices for free.
Worked answer
- Put what you know into .
- .
- .
- So .
Answer: edges
Two of the three counts were enough. The rule handed over the third.
Worked answer
- The four triangles meet at one point on top.
- Underneath they leave a square hole, so the square base is missing.
- With that base the pyramid has faces, edges and vertices.
- Check the rule: .
Answer: the square base
Counting only the triangles leaves the pyramid with no floor.
Traps
- Naming a solid by counting its vertices instead of its faces. A Platonic solid takes its name from its number of faces, not its corners.
- Counting only the faces you can see in a drawing of a solid. A drawing hides some faces. Picture the whole object, or use its net, to count them all.
- Putting the number of edges in the spot where vertices belong in . Keep , and straight: vertices, edges, then faces, in that order.
- Assuming any arrangement of the right faces makes a valid net. Picture folding the net up. The faces must meet edge to edge without overlapping.